An important tool in understanding the representation theory for a Lie algebra \(\mathfrak {g}\)  is the study of restrictions of an irreducible representation \(\rho \)  for \(\mathfrak {g}\)  to a Lie subalgebra \(\mathfrak {g}^{\prime } \subset \mathfrak {g}\) . This so-called restricted representation will in general no longer be irreducible as it decomposes into a direct sum of irreducible representations for \(\mathfrak {g}^{\prime }\) . The algebraic tool which allows us to describe which \(\mathfrak {g}^{\prime }\) -irreducible summands show up and what their multiplicities are is called a branching rule; see e.g. Goodman's and Wallach's book [1]} for classical branching rules. These abstract rules get a physical meaning in the theory of symmetry breaking, see for instance [2]} and the references therein for more details on symmetry breaking and its applications in physics (e.g. splitting of degenerate energy levels into multiplets in quantum mechanics). Moreover, the way in which \(\mathfrak {g}^{\prime }\) -irreducible representations appear in \(\mathfrak {g}\)  often gives rise to special functions (the Gegenbauer polynomials appearing in harmonic analysis are an important example).
